Question

Cho các biểu thức:

A=\(\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}}{x+\sqrt{x}+1}+\frac{1}{1-\sqrt{x}}\) và B= \(\frac{\sqrt{x}-1}{2}\) với x≥0 x≠1

a) Rút gọn biểu thức A

b)Tìm giá trị lớn nhất của biểu thức P=\(\frac{A}{B}\)

Answer 1

\(a,A=\frac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(\Rightarrow A=\frac{x+2+\sqrt{x}\left(\sqrt{x}-1\right)+x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(\Rightarrow A=\frac{x+2+x-\sqrt{x}+x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(\Rightarrow\frac{3x+3}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(b,Tacó:P=\frac{A}{B}=\frac{3x+3}{2\left(x+\sqrt{x}+1\right)}\)

\(\Rightarrow P=\frac{3}{2}.\frac{x+1}{x+\sqrt{x}+1}\)

\(\Rightarrow P=\frac{3}{2}.\frac{x+1}{x+1+\sqrt{x}}\)

\(\Rightarrow P=\frac{3}{2}.\left(1-\frac{\sqrt{x}}{x+1+\sqrt{x}}\right)\)

\(\Rightarrow P\le\frac{3}{2}.\left(1-0\right)\)

\(\Rightarrow P\le\frac{3}{2}\)

\(\Rightarrow Max_P=\frac{3}{2}\)